In do Carmo's book on Differential Geometry of Curves and Surfaces, the proof of theorema egregium, that the Gauss curvature of a surface immersed in $\mathbb{R}^3$ is invariant under local isometries, requires the surface to be of class at least $C^3$.
What happens if the surface is of class $C^2$ only ? The Gauss curvature can still be defined but is it invariant under local isometries ?
Best,
Ryan
This question depends strongly on the regularity of the local isometries. If the surface is $C^2$ and there's a local isometry of class $C^2$, it will indeed preserve the Gaussian curvature. This is shown in the paper "On the Fundamental Equations of Differential Geometry" by Philip Hartman and Aurel Wintner. Nash's isometric $C^1$-embedding theorem shows that curvature occurs no longer as an obstruction to $C^1$-isometries. There are embeddings of $S^2$ into arbitrarily small portions of $\mathbb R^3$ which is forbidden if curvature makes sense. $C^2$-isometric embeddings of $S^2$ into $\mathbb R^3$ are unique up to the action of the Euclidean group in the target. This however is a non-trivial regularity phenomenon.
A side note: The rigidity stated for $C^2$-embeddings of the sphere holds furthermore for $C^{1,\alpha}$-embeddings provided $\alpha>2/3$, while the Nash-type flexibility remains true if for some small $\alpha>0$. The critical $0<\alpha<1$ is unknown.